Factor the following expression: $-4$ $x^2+$ $13$ $x+$ $12$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-4)}{(12)} &=& -48 \\ {a} + {b} &=& & & {13} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-48$ and add them together. Remember, since $-48$ is negative, one of the factors must be negative. The factors that add up to ${13}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-3}$ and ${b}$ is ${16}$ $ \begin{eqnarray} {ab} &=& ({-3})({16}) &=& -48 \\ {a} + {b} &=& {-3} + {16} &=& 13 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-4}x^2 {-3}x +{16}x +{12} $ Group the terms so that there is a common factor in each group: $ ({-4}x^2 {-3}x) + ({16}x +{12}) $ Factor out the common factors: $ x(-4x - 3) - 4(-4x - 3) $ Notice how $(-4x - 3)$ has become a common factor. Factor this out to find the answer. $(-4x - 3)(x - 4)$